Question

If the sum of hypotenuse and side of right angeled triangle is given. then show that area of the triangle is maximum , when angle betwee them is 60

Answer 1

Let the hypotenuse of the right triangle be x, and the height be y

Hence its base is x2−y2−−−−−−√ by applying phythagorous theorem.

Hence its are = 12×base×height

Area = 12×x2−y2−−−−−−√×y

But it is given x+y=p(say)

Substituting this in the area we get

Area = 12×(p−y)2−−−−−−−√−y2×y

12yp2+y2−2py−y2−−−−−−−−−−−−−−−√

=12yp2−2py−−−−−−−√

Squaring on both the sides we get

(Area)2=14y2(p2−2py)

i.e., A=14y2(p2−2py)

=14p2y2=12py3

For maximum or miniumu area

dydA=0

Here the area of the triangle is maximum when x=2p3andy=p3

cosθ=yx

=p32p3

∴cosθ=12

⇒θ=π3 or 60∘

Hence the area is maximum if the angle between the hypotenuse and the side is 60∘